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FRM Part 1 · Worked study notes

VaR Mapping and Model Limitations

VaR estimates a loss quantile for a specified horizon and confidence level. Map positions to the factors that drive their value, then choose a distribution or simulation method. A VaR number is conditional on that model, not a maximum possible loss.

The method at a glance

  • 01Parametric

    How it works: Combine sensitivities, factor volatilities and correlations.

    Watch for: Applying a linear normal approximation to a strongly nonlinear payoff.

  • 02Historical

    How it works: Revalue today’s portfolio under selected historical shocks.

    Watch for: Assuming the sample contains every relevant stress.

  • 03Monte Carlo

    How it works: Simulate joint factor moves and revalue the portfolio.

    Watch for: Confusing more simulations with better model assumptions.

Original worked example

One-day normal VaR

A linear portfolio is worth 10 million with daily return volatility of 1%. Assume zero mean and a normal one-day return; use z = 2.326 for 99%.

  1. STEP 1

    Daily money volatility = 10,000,000 × 0.01 = 100,000.

  2. STEP 2

    99% VaR ≈ 2.326 × 100,000 = 232,600.

  3. STEP 3

    The model assigns about 1% probability to a loss exceeding this threshold.

VaR is approximately 232,600. It does not estimate the average loss beyond the threshold or cap the loss.

Illustrative assumptions and calculations for learning. Not observed market data or an official exam item.

Apply the method and check its limits

Before you start: Normal quantiles, volatility, covariance and a consistent loss sign convention.

Map factor changes into a loss distribution

σ²L ≈ bᵀΣb; VaRα ≈ μL + zα σL

Here b contains signed loss sensitivities and Σ is the covariance matrix of matching factor changes. If sensitivities are currency per basis point, the rate-shock covariance must use basis points squared. For a linear portfolio with normal losses, the formula gives a parametric quantile. Options can require second-order terms or full revaluation.

Know what the reported quantile leaves out

A 99% one-day VaR does not state the maximum loss or the size of losses beyond its cutoff. Square-root-of-time scaling needs assumptions about dependence and variance; it is not a rule for every portfolio. Historical simulation depends on the chosen window, while normal VaR can understate risk when losses have fatter tails or changing volatility.

What breaks a square-root-of-time shortcut?

Scaling one-day VaR by √10 assumes a suitable stable distribution, independent increments and effectively unchanged exposure. Serial dependence, jumps, volatility changes and options can violate those assumptions. Specify whether P&L is actual or hypothetical and whether positions can be rebalanced over the horizon before comparing two reported VaRs.

Work through the topic

Follow a VaR estimate from positions to simulated losses, then identify the assumptions that can make its apparent precision misleading.

Define the loss before choosing the quantile

Value at Risk is a quantile of a specified loss distribution. Start with a portfolio, valuation time, holding period and confidence level. Under a positive-loss convention, a one-day 99% VaR of 2 million describes a threshold that the model expects losses to exceed on about 1% of comparable days. It is neither the largest possible loss nor the average loss on those adverse days. These distinctions matter even when a calculator produces the correct number.

For historical simulation, take a defined set of past risk-factor changes, apply each to today’s positions and revalue. Sort the resulting hypothetical losses, then select a quantile using a stated convention. Do not substitute the historical profit and loss of a portfolio whose holdings changed each day. That series mixes market changes with changes in the investment strategy and may answer a different question.

Suppose 100 equally weighted scenarios produce losses ordered from smallest to largest. Under the nearest-rank convention, the 95% VaR is the 95th ordered observation. The five observations above it determine how severe the upper tail is, but their magnitude does not change that particular order statistic unless the ordering changes. With only 100 observations, the extreme tail is represented by very little evidence. Interpolation and weighting rules can also change the reported quantile.

Map the economic exposure before estimating volatility

Mapping translates contractual positions into a set of modeled drivers. A foreign-currency corporate bond can depend on several yield-curve tenors, a credit spread and an exchange rate. Mapping it only to a broad government-bond index omits important channels. The data may be clean and the covariance calculation flawless while the risk estimate remains incomplete because the chosen factors do not represent the instrument.

In an original linear example, a position gains or loses 100,000 for a one-unit factor move. The factor’s daily standard deviation is 0.2 units, giving a position standard deviation of 20,000. Assuming zero mean and a normal loss distribution, 99% VaR is approximately 2.326 × 20,000 = 46,520. This estimate depends on linear sensitivity, the volatility estimate, the distribution and the horizon. It is not a general valuation formula for options.

Adding factors requires their joint behavior. Two separately measured risks cannot automatically be added as if their worst outcomes always coincide, or diversified as if they were independent. For a linear portfolio, covariance aggregates factor exposures under the selected model. For nonlinear positions, revaluation or additional sensitivity terms may be needed. Keep factor selection separate from the statistical assumptions used after selection.

Where a VaR estimate can become incomplete

Position mapping
Original example of a weakness: Corporate bond represented only by a government rate
Useful challenge: Where are spread and currency changes?
Scenario generation
Original example of a weakness: Quiet sample used after a regime change
Useful challenge: How does a stressed sample change the result?
Valuation
Original example of a weakness: Option represented by fixed delta
Useful challenge: Does full repricing change the tail?
Reporting
Original example of a weakness: Threshold presented as maximum loss
Useful challenge: What happens beyond the quantile?

Challenge time scaling and model performance

Square-root-of-time scaling assumes a suitable aggregation structure, commonly independent, identically distributed increments with stable variance and an approximately linear exposure. Multiplying one-day volatility by the square root of ten does not establish that a ten-day liquidation will occur under unchanged market depth or position sensitivity. Serial dependence, changing volatility, jumps and changing positions can make that shortcut unreliable.

A lower reported VaR can reflect a smaller position, a quieter estimation window or an omitted risk. Reconcile the change before praising it. Hold today’s positions constant and compare model versions, then hold the model constant and compare positions. Those comparisons help separate business decisions from methodological changes. They do not remove model uncertainty, but they make its sources more visible.

Use backtesting, stress testing and exposure reviews for different questions. Backtesting compares forecasts with subsequent outcomes under a consistent profit-and-loss definition. Stress testing examines specified adverse conditions, including conditions poorly represented in the estimation sample. Neither a successful backtest nor a plausible stress scenario proves completeness. The Part II backtesting lesson develops exception counts and clustering; this page establishes what must be specified before those tests mean anything.

Can two models have the same VaR and different tail risk?

Yes. They can share a quantile while assigning very different sizes to losses beyond it. Compare expected shortfall, scenario losses and the mechanisms that produce the tail. A single quantile cannot identify the whole distribution.

Should a small VaR always lead to a larger position?

No. Review concentration, liquidity, nonlinear exposure, data coverage and the organization’s constraints first. VaR is an input to a decision, not authorization to trade. A small estimate based on missing factors can be particularly misleading.

Try a changed assumption

If position size doubles but volatility and the linear exposure assumptions stay fixed, what happens to the 232,600 VaR example?

Show answer and reasoning

It doubles to 465,200. That scaling is justified by the stated linear setup. It does not imply that market liquidity or transaction costs also scale linearly.

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Can I add individual VaRs?

Not generally to get an exact portfolio VaR. Dependence matters, and VaR is not subadditive for all loss distributions. Aggregate the joint losses or use a suitable portfolio model.

Sources and scope

Links checked 2026-10-10. GARP establishes exam scope; the technical references support the methods identified below. These are selected study notes and original examples, not complete curriculum coverage or official exam questions.

Prepared by MiloPrep editorial team. MiloPrep is independent of GARP. References do not imply endorsement or independent review of these notes.